# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Bifurcation of a homoclinic orbit with a saddle-node equilibrium. (English) Zbl 0727.58026

One studies the codimension two unfolding of an orbit ${{\Gamma }}_{0}$ homoclinic to a saddle-node equilibrium in ${ℝ}^{d}$. The eigenvalues of the equilibrium are supposed to be positive and negative real parts, except for a simple eigenvalue ${\lambda }_{0}=0$. Several new methods are employed. The bifurcation diagram is obtained by the bifurcation function which is derived by the method of Lyapunov-Schmidt decomposition. The idea of exponential trichotomy is employed to study the linearized equation around ${{\Gamma }}_{0}·$

The method of cross sections and Poincaré mappings are used to study the bifurcation of the periodic orbits from ${{\Gamma }}_{0}·$

Under the Poincaré mapping, submanifolds of certain type, called the u- slices can be found to be fixed and hence bifurcation of periodic orbits is proved.

##### MSC:
 37G99 Local and nonlocal bifurcation theory 37-99 Dynamic systems and ergodic theory (MSC2000) 34C45 Invariant manifolds (ODE) 37C70 Attractors and repellers, topological structure